Method of determining net reservoir thickness

ABSTRACT

The methods of the present invention determine the net and gross thickness of reservoir beds, taking into account the internal structure of reservoir beds, using P-P, and/or S-S and/or P-S seismic data acquired in 2-d or 3-d.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No. 60/100,019, filed Sep. 11, 1998.

BACKGROUND OF THE INVENTION

This invention relates generally to beds, or reservoir layers, and in particular to methods for determining the net and gross thickness of a bed. Hydrocarbon reservoirs in general are not perfect blocky beds, but rather have internal structure in the form of embedded layers of non-reservoir or poor reservoir quality material. However, existing methods for determining gross or net reservoir thickness of these reservoirs ignore or do not accurately consider the internal structure. Thus, those methods have neglected to include effects associated with the internal structure of beds, in the determination of the net and gross thickness of beds.

SUMMARY OF THE INVENTION

The method of the present invention, for the determination of the net and gross thickness of reservoir beds, solves the difficulties of other existing methods, by providing a method that takes into account the internal structure of reservoir beds.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an illustration of the band pass filters, and the outputs generated by those filters, used in the method of the present invention.

FIG. 2 is a cross-sectional illustration of different types of beds which are analyzed using the method of the present invention.

FIG. 3 is an illustration of an amplitude signature of a blocky bed, with an embedded reservoir layer.

FIG. 4 is an illustration of an amplitude signature of the bed of FIG. 3, with another, thinner bed embedded within it.

FIG. 5 is an illustration of the amplitude signatures of the beds of FIGS. 3 and 4, with a curve showing the effect of the embedded bed.

FIG. 6 is an illustration of an amplitude signature of a blocky bed, with embedded non-reservoir layers.

DETAILED DESCRIPTION OF THE INVENTION

The methods of the present invention, for determining the gross and net thickness of a reservoir bed, or layer, have in common the following steps. First, as shown in FIG. 1, the seismic data is filtered by a series of band pass filters, 10, 12, 14, each of which has a fixed lower frequency f_(L) but with progressively higher upper frequency f_(U). Next, the amplitudes are extracted on a lobe, or sample by sample, basis from filtered seismic traces 16, 18, 20, or from the energy envelope derived from the filtered seismic traces. These amplitude values A_(i) and the corresponding frequencies f_(Ui), which characterize the filter applied, make up the “amplitude signature” which, as described below, is used to determine the gross and net thickness.

Referring now to FIG. 2, a variety of types of information is extracted from the amplitude signatures related to the properties of the beds 30, 32 being imaged. These extracted properties will be referred to as Frequency Dependant Attributes or FDA's. The bed geometry described applies for P-P, P-S and S-S data. These time thicknesses Δt and δt_(i) are related to each other and to ΔZ as shown below in equations. For all of the methods described, the relationships between the various thicknesses are defined and related to each other as:

ΔZ=spatial thickness

Δt=gross reservoir thickness in two-way travel time

δt_(i)=the two-way time thickness of the embedded ith layer; and

Δt_(net)=Δt−Σδt_(i) the net reservoir thickness in two-way travel time

For the case of a single embedded bed, the net thickness is:

Δt_(net)=Δt−δt=the net reservoir thickness in two-way travel time.

The amplitude signature for a blocky reservoir bed is: $\begin{matrix} {{{AMP}\left( f_{U} \right)} = {\frac{\Delta \quad {Zp}}{2{Zp}}\left( {1 - \frac{{SIN}\left( {{\alpha\Delta}\quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right)}} & {{Equation}\quad 1} \end{matrix}$

where $\begin{matrix} \begin{matrix} {{\frac{\Delta \quad {Zp}}{Zp} = {{the}\quad {relative}\quad {impedance}\quad {contrast}\quad {for}\quad {the}\quad {reservoir}}};} \\ {{\Delta \quad t} = \text{gross~~reservoir~~thickness~~in~~two-way~~travel~~time;}} \end{matrix} \\ \begin{matrix} {f_{U} = \text{the~~upper~~frequency~~of~~the~~filter~~used; and}} \\ {\alpha = {2{\pi.}}} \end{matrix} \end{matrix}$

FIG. 3 illustrates the amplitude signature 34 given by Equation 1, for the case of Δt=.02 seconds.

The amplitude signature for a blocky reservoir bed with one or more smaller non-reservoir layers embedded in it is approximately: $\begin{matrix} {{{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\sum\limits_{i}\left( {1 - \frac{{SIN}\left( {\alpha \quad \delta \quad t_{i}f_{U}} \right)}{\alpha \quad \delta \quad t_{i}f_{U}}} \right)}} \right\rbrack}} & {{Equation}\quad 2} \end{matrix}$

where the time reference is the position in the trace where the amplitude is an extrema (“Extrema” means the largest value, regardless of the sign, that is, regardless of whether it is positive or negative), and where $\begin{matrix} \begin{matrix} \begin{matrix} {{\frac{\Delta \quad {Zp}}{Zp} = {{the}\quad {relative}\quad {impedance}\quad {contrast}\quad {for}\quad {the}\quad {reservoir}}};} \\ {{\Delta \quad t} = \text{the~~two-way~~time~~thickness~~with~~the~~reservoir~~layer;}} \end{matrix} \\ {{\delta \quad t_{i}} = \text{the~~two-way~~time~~thickness~~of~~the~~ith~~layer;}} \end{matrix} \\ \begin{matrix} {f_{U} = \text{the~~upper~~frequency~~of~~the~~filter; and}} \\ {\alpha = {2{\pi.}}} \end{matrix} \end{matrix}$

FIG. 4 illustrates the amplitude signature 36 for the bed 32, given by Equation 2, for the case of Δt=0.02 seconds and for a single embedded layer δt_(i)=0.005.

Equation 2 reveals that the effect of one or more non-reservoir beds embedded in a thicker reservoir layer is to produce the expected amplitude response of the gross reservoir (see curve 40 in FIG. 5, the black curve), with the polarity reversed amplitude response curve of the unresolved embedded non-reservoir bed (see curve 42FIG. 5, red curve). The net effect is to produce an amplitude decay with increasing frequency (see curve 44, FIG. 5, green curve). From this amplitude decay with frequency effect, curve 44, the net bed thickness is determined as described below.

For the case of a single embedded layer of thickness δt: $\begin{matrix} {{{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - \left( {1 - \frac{{SIN}\left( {\alpha \quad \delta \quad {tf}_{U}} \right)}{\alpha \quad \delta \quad {tf}_{U}}} \right)} \right\rbrack}} & {{Equation}\quad 3} \end{matrix}$

where the time reference is the position in the trace where the amplitude is an extrema, and where $\begin{matrix} \begin{matrix} \begin{matrix} {{\frac{\Delta \quad {Zp}}{Zp} = {{the}\quad {relative}\quad {impedance}\quad {contrast}\quad {for}\quad {the}\quad {reservoir}}};} \\ {{\Delta \quad t} = \text{the~~two-way~~time~~thickness~~with~~the~~reservoir~~layer;}} \end{matrix} \\ {{\delta \quad t_{i}} = \text{the~~two-way~~time~~thickness~~of~~the~~embedded~~layer;}} \end{matrix} \\ \begin{matrix} {f_{U} = \text{the~~upper~~frequency~~of~~the~~filter; and}} \\ {\alpha = {2{\pi.}}} \end{matrix} \end{matrix}$

There are situations where there can be distinct advantages obtained by approximating Equations 2 and 3 to give Equations 4 and 5, which take advantage of the fact that the thickness of the embedded beds is less than the gross thickness Δt. Thus, Equations 4 and 5 are used in a variety of ways as shown below to solve for the gross and net thickness of the bed. Specifically, if an estimate is made for Δt, then either Σδ_(i) ² is found by a linear fitting process (method 3 and 4), or it can be directly solved for Σδ_(i) ² and Δt as described below (methods 8 and 9).

The amplitude signature for a blocky reservoir bed with one or more smaller non-reservoir layers embedded in it is approximately: $\begin{matrix} {{{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\sum\limits_{i}{\frac{2\pi^{2}}{3}\left( {\delta \quad t_{i}f_{U}} \right)^{2}}}} \right\rbrack}} & {{Equation}\quad 4} \end{matrix}$

where the time reference is the position in the trace where the amplitude is an extrema, and where $\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} {{\frac{\Delta \quad {Zp}}{Zp} = {{the}\quad {relative}\quad {impedance}\quad {contrast}\quad {for}\quad {the}\quad {reservoir}}};} \\ {{\Delta \quad t} = \text{the~~two-way~~time~~thickness~~with~~the~~reservoir~~layer;}} \end{matrix} \\ {{\delta \quad t_{i}} = \text{the~~two-way~~time~~thickness~~of~~the~~ith~~layer;}} \end{matrix} \\ {f_{U} = \text{the~~upper~~frequency~~of~~the~~filter; and}} \end{matrix} \\ {\alpha = {2{\pi.}}} \end{matrix}$

The amplitude signature for a blocky reservoir bed with a single non-reservoir layer embedded in it, of thickness δt, is approximately: $\begin{matrix} {{{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}}} \right\rbrack}} & {{Equation}\quad 5} \end{matrix}$

where the time reference is the position in the trace where the amplitude is an extrema, and where $\begin{matrix} \begin{matrix} \begin{matrix} {{\frac{\Delta \quad {Zp}}{Zp} = {{the}\quad {relative}\quad {impedance}\quad {contrast}\quad {for}\quad {the}\quad {reservoir}}};} \\ {{\Delta \quad t} = \text{the~~two-way~~time~~thickness~~with~~the~~reservoir~~layer;}} \end{matrix} \\ {{\delta \quad t_{i}} = \text{the~~two-way~~time~~thickness~~of~~the~~embedded~~layer;}} \end{matrix} \\ \begin{matrix} {f_{U} = \text{the~~upper~~frequency~~of~~the~~filter; and}} \\ {\alpha = {2{\pi.}}} \end{matrix} \end{matrix}$

In the case where the accumulated thickness of the embedded layers Σδt_(i) approaches the thickness Δt, then Equation 2 becomes: $\begin{matrix} {{{AMP}\left( f_{U} \right)} \cong {{\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\quad\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\sum\limits_{i}{\frac{2\pi^{2}}{3}\left( {\delta \quad t_{i}f_{U}} \right)^{2}}} + {\sum\limits_{i}{\frac{2\pi^{4}}{15}\left( {\delta \quad t_{i}f_{U}} \right)^{4}}}} \right\rbrack}}} & {{Equation}\quad 6} \end{matrix}$

where the time reference is the position in the trace where the amplitude is an extrema, and where $\begin{matrix} \begin{matrix} \begin{matrix} {{\frac{\Delta \quad {Zp}}{Zp} = {{the}\quad {relative}\quad {impedance}\quad {contrast}\quad {for}\quad {the}\quad {reservoir}}};} \\ {{\Delta \quad t} = \text{the~~two-way~~time~~thickness~~with~~the~~reservoir~~layer;}} \end{matrix} \\ {{\delta \quad t_{i}} = \text{the~~two-way~~time~~thickness~~of~~the~~embedded~~layer;}} \end{matrix} \\ \begin{matrix} {f_{U} = \text{the~~upper~~frequency~~of~~the~~filter; and}} \\ {\alpha = {2{\pi.}}} \end{matrix} \end{matrix}$

The amplitude signature for a blocky reservoir bed with a single non-reservoir layer embedded in it, of thickness δt, is approximately given by: $\begin{matrix} {{{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}} + {\frac{2\pi^{4}}{15}\left( {\delta \quad {tf}_{U}} \right)^{4}}} \right\rbrack}} & {{Equation}\quad 7} \end{matrix}$

where the time reference is the position in the trace where the amplitude is an extrema, and where $\begin{matrix} \begin{matrix} \begin{matrix} {{\frac{\Delta \quad {Zp}}{Zp} = {{the}\quad {relative}\quad {impedance}\quad {contrast}\quad {for}\quad {the}\quad {reservoir}}};} \\ {{\Delta \quad t} = \text{the~~two-way~~time~~thickness~~with~~the~~reservoir~~layer;}} \end{matrix} \\ {{\delta \quad t_{i}} = \text{the~~two-way~~time~~thickness~~of~~the~~embedded~~layer;}} \end{matrix} \\ \begin{matrix} {f_{U} = \text{the~~upper~~frequency~~of~~the~~filter; and}} \\ {\alpha = {2{\pi.}}} \end{matrix} \end{matrix}$

FIG. 5 illustrates the accuracy of the approximations (curves 46 and 48) for the case where Δt=.02 seconds, and there is only a single embedded layer of thickness δt=.005 seconds.

Referring now to FIG. 6, given two measured amplitudes and their corresponding frequencies, Ax, fx and Ah, fh, the gross and net thicknesses are calculated using the following equations: $\begin{matrix} {{\Delta \quad t} = {\frac{5}{7{fx}} - ɛ}} & {{Equation}\quad 8} \end{matrix}$

where ε=2.14*ƒx* Σδt_(i) ² for the case of more than one embedded layers or

for the case of a single embedded layer

δ=2.14*ƒx*δt²

and $\begin{matrix} {{\sum{\delta \quad t}} = {(1.013)\sqrt{\frac{{C_{1}f_{x}{{SIN}\left\lbrack \frac{10\pi \quad f_{h}}{7f_{x}} \right\rbrack}} + {C_{2}f_{h}\frac{A_{h}}{A_{x}}} - {C_{3}f_{h}}}{{C_{4}f_{h}f_{x}^{2}} - {C_{5}f_{h}^{3}} - {C_{6}f_{x}^{3}{{SIN}\left\lbrack \frac{10\pi \quad f_{h}}{7f_{x}} \right\rbrack}}}}}} & {{Equation}\quad 9} \end{matrix}$

or

for the case of a single embedded layer ${\delta \quad t} = {(1.013)\sqrt{\frac{{C_{1}f_{x}{{SIN}\left\lbrack \frac{10\pi \quad f_{h}}{7f_{x}} \right\rbrack}} + {C_{2}f_{h}\frac{A_{h}}{A_{x}}} - {C_{3}f_{h}}}{{C_{4}f_{h}f_{x}^{2}} - {C_{5}f_{h}^{3}} - {C_{6}f_{x}^{3}{{SIN}\left\lbrack \frac{10\pi \quad f_{h}}{7f_{x}} \right\rbrack}}}}}$

where

C₁=1.299

C₂=7.10

C₃=5.83

C₄=32.36

C₅=39.39

C₆=7.211

δt is in seconds.

In all of the discussion above concerning bed thickness, whether net or gross, δT_(2WAY) is the two-way travel time for the bed. These travel times are for P-P, P-S, or S-S data types. The spatial thickness ΔZ of the layer is given, for each of these cases by: $\begin{matrix} \begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot \left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right)}} \end{matrix} & {{{Equations}\quad 10},\quad {11\quad {and}\quad 12}} \end{matrix}$

where:

ΔT_(PP-2WAY)=the 2-way P-P travel time;

ΔT_(SS-2WAY)=the 2-way S-S travel time;

ΔT_(PS-2WAY)=the 2-way P-DOWN S-UP travel time; and

ΔZ=the thickness of the layer.

There are a number of methods for quantitatively determining the gross and net bed thicknesses of a reservoir layer using the above equations. They include:

1) Perform a nonlinear fit, using Equation 2, to amplitude and fu_(i) pairs to produce optimal data consistent values for Δt, δt_(i) and ΔZp/Zp. Δt, δt_(i) and ΔZp/Zp are the data derived FDA's for this method.

2) Perform a nonlinear fit, using Equation 3, to amplitude and f_(Ui) pairs to produce optimal data consistent values for Δt, δt and ΔZp/Zp for the case where only a single embedded layer is present Δt, δt and ΔZp/Zp are the data derived FDA's for this method.

3) Derive Equation 4 by simplifying Equation 2 by taking advantage of the fact that the total thickness of the layers t_(i) are small compared with Δ. The value for Δt is initially estimated by finding the frequency for which the amplitude is at its maximum ƒx (see FIG. 6) and then substituting Δt=1(1.4*ƒx). Next, perform a linear fit, using Equation 4, to amplitude and fu_(i) pairs to produce optimal data consistent values for (Σδt_(i) ²) and ΔZp/Zp. With this initial estimate of Σδt_(i) ², Equation 8 is used to improve the initial estimate made for the gross thickness Δt. This improved value for Δt is substituted into Equation 4, and a new fit performed, resulting in an improved estimate for Σδt_(i) ². FIG. 5 illustrates the accuracy of this approach. The red curve, curve 42, represents the exact response while the approximate form, Equation 4, is the cyan curve, curve 48. Δt, δt_(i) and ΔZp/Zp are the data derived FDA's for this method.

4) Simplify Equation 3 by taking advantage of the fact that the thickness of the embedded layer δt is small compared with Δt for the case where only a single embedded layer is present. This gives Equation 5. The value for Δt is initially estimated by finding the frequency for which the amplitude is at its maximum ƒx (see FIG. 6) and then substituting Δt=1/(1.4*ƒx). Then a linear fit to the data is made, using Equation 5, to amplitude and f_(Ui) pairs to produce optimal data consistent values for (δt²) and ΔZp/Zp. With this initial estimate of δt^(2,) Equation 8 is used to improve the initial estimate made for the gross thickness Δt. This improved value for Δt is substituted into Equation 5, and a new fit performed, resulting in an improved estimate for δt². Δt, δt and ΔZp/Zp are the data derived FDA's for this method.

5) In the case where the thickness of the embedded layer approaches the reservoir thickness, then an improved approximation to Equations 4 and 5 are 6 and 7 respectively. These equations result from simplifying Equation 2 by taking advantage of the fact that the total thickness of the layers δt_(i) are small compared with Δt. This gives Equation 6. The value for Δt is initially estimated by finding the frequency for which the amplitude is at a maximum ƒx (see FIG. 6) and then substituting Δt=1(1.4*ƒx). Perform a non-linear fit, using Equation 6, to amplitude and f_(Ui) pairs to produce optimal data consistent values for (Σδt_(i) ²) and ΔZp/Zp. With this initial estimate of Σδt_(i) ², Equation 8 is used to improve the initial estimate made for the gross thickness Δt. This improved value for Δt is substituted into Equation 4, and a new fit is performed, resulting in an improved estimate for Σδt_(i) ². FIG. 5 illustrates the accuracy of this approach. The red curve (curve 42) represents the exact response, while the approximate form, indicated in Equation 6, is the dark blue curve, curve 46. Δt, δt_(i) and ΔZp/Zp are the data derived FDA's for this method.

6) For the case where only a single embedded layer is present it is possible to simplify Equation 6 by taking advantage of the fact that the thickness of the embedded layer δt is somewhat small compared with Δt. This gives Equation 7. The value for Δt is initially estimated by finding the frequency for which the amplitude is at a maximum ƒx (see FIG. 6), and then substituting Δt=1/(1.4*ƒx). Then a non-linear fit to the data is made, using Equation 7, to amplitude and fu pairs to produce optimal data consistent values for (δt²) and ΔZp/Zp. With this initial estimate of δt², Equation 8 is used to improve the initial estimate made for the gross thickness Δt. This improved value for Δt is substituted into Equation 7, and a new fit is performed, resulting in an improved estimate for δt². Δt, δt and ΔZp/Zp are the data derived FDA's for this method.

The final 2 methods do not require any type of linear or nonlinear curve fitting.

7) This method assumes the total thickness of the embedded layers is less than the gross bed thickness. In this case the gross and embedded bed thicknesses are solved for by using Equation 4. The result is Equations 8 and 9. All that is required are two amplitude measurements, Ax and Ah, and their corresponding frequencies, fx and fh, where the first measurement is made where the amplitude is at its maximum (Ax,fx) and the second is for a frequency at least 10% higher than the frequency where the amplitude is at its maximum. These are Ah and fh. This arrangement is shown in FIG. 6. The value for Σδt_(i) is given by Equation 9, while the value for the gross thickness is given by Equation 8. Δt, and Σδt_(i) are the data derived FDA's for this method.

8) This method assumes the total thickness of the embedded layers is less than the gross bed thickness, and that there is only a single embedded layer present. In this case the gross and embedded bed thicknesses are solved for by using Equation 5. The result is Equations 8 and 9. All that is required are two amplitude measurements, Ax and Ah, and their corresponding frequencies, fx and fh, where the first measurement is made where the amplitude is at its maximum (Ax, fx) and the second is for a frequency at least 10% higher than the frequency where the amplitude is at its maximum. These are Ah and fh. This arrangement is shown in FIG. 6. The value for δt is given by Equation 9, while the value for the gross thickness is given by Equation 8. Δt, and δt_(i) are the data derived FDA's for this method.

In summary, the following methods of exploration and evaluation have been shown:

1. A method of determining the net thickness of a reservoir bed, comprising the following steps:

a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces;

b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui);

a) determining the gross reservoir thickness Δt and the thickness of the non-reservoir material Σδ₁, using the extracted amplitude frequency pairs A_(i) and the f_(Ui),; and

d) determining the net thickness of the reservoir by $\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad {\underset{i}{t}.}}}}$

2. A method of determining the net thickness of a reservoir bed, comprising the following steps:

a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces;

b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui);

c) fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, using nonlinear fitting methods, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\sum\limits_{i}\left( {1 - \frac{{SIN}\left( {\alpha \quad \delta \quad t_{i}f_{U}} \right)}{\alpha \quad \delta \quad t_{i}f_{U}}} \right)}} \right\rbrack}$

to generate fit parameters Δt, Εδt_(i), and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

d) using the fit parameters Δt and Σδ_(I), determining the net thickness of the reservoir by using ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad {\underset{i}{t}.}}}}},$

and

e) using the gross thickness Δt and the net thickness of the reservoir ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad {\underset{i}{t}.}}}}},$

determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot \left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right)}} \end{matrix}$

depending on whether the data type is P-P, S-S or P-S respectively.

3. A method of determining the net thickness of a reservoir bed, comprising the following steps:

a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui) to generate filtered seismic traces;

b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui);

a) in the case where only a single embedded layer of thickness δt is present, using nonlinear fitting methods, fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - \left( {1 - \frac{{SIN}\left( {\alpha \quad \delta \quad {tf}_{U}} \right)}{\alpha \quad \delta \quad {tf}_{U}}} \right)} \right\rbrack}$

to generate fit parameters Δt, δt, and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

d) using the fit parameters Δt and δt, determining the net thickness of the reservoir by using Δ_(iNET)=Δt−δt; and

f) using the gross thickness Δt and the net thickness of the reservoir Δ_(iNET)=Δt−δt, determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot \left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right)}} \end{matrix}$

depending on whether the data type is P-P, S-S or P-S respectively.

4. A method of determining the net thickness of a reservoir bed, comprising the following steps:

a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces;

b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui);

c) determining the frequency fx at which the amplitudes are at their maximum value by analyzing the amplitude frequency pairs A_(i) and f_(I);

d) using linear fitting methods, and using Δt=1/(1.4*fx), fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\sum\limits_{i}{\frac{2\pi^{2}}{3}\left( {\delta \quad t_{i}f_{U}} \right)^{2}}}} \right\rbrack}$

to generate fit parameters Δt, Σδ_(i), and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

e) substituting the fit parameter Σδt_(i) into ${\Delta \quad t} = {{\frac{5}{7{fx}} - {ɛ\quad {where}\quad ɛ}} = {2.14*{fx}*{\sum{\delta \quad t_{i}^{2}}}}}$

to produce an improved estimate of Δt;

f) fitting the amplitude and frequency pairs Δ_(i) and f_(Ui) to the following equation, using linear fitting methods, and using the improved estimate of Δt, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\sum\limits_{i}{\frac{2\pi^{2}}{3}\left( {\delta \quad t_{i}f_{U}} \right)^{2}}}} \right\rbrack}$

to generate new fit parameters Δt, Σδt_(i), and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

g) using the new fit parameters Δt and Σδt_(i), determining the net thickness of the reservoir by ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad \underset{i}{t}}}}};$

and

g) using the gross thickness Δt and the net thickness of the reservoir ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad \underset{i}{t}}}}},$

determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot \left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right)}} \end{matrix}$

depending on whether the data type is P-P, S-S or P-S respectively.

5. A method of determining the net thickness of a reservoir bed, comprising the following steps:

a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces;

b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui);

a) in the case where only a single embedded layer of thickness δt is present, determining the frequency fx at which the amplitudes are at their maximum value by analyzing the amplitude frequency pairs A_(i) and f_(Ui);

d) using linear fitting methods, and using Δt=1/(1.4*fx), fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}}} \right\rbrack}$

to generate fit parameters Δt, δ, and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

e) substituting the fit parameter δt into the following equation, ${{\Delta \quad t} = {\frac{5}{7{fx}} - {ɛ\quad {where}}}}\quad$

ε=2.14*ƒx*δt²

to produce an improved estimate of Δt;

f) using the improved estimate of Δt, and using linear fitting methods, fitting the amplitude and frequency pairs Δ_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}}} \right\rbrack}$

to generate new fit parameters Δt, δt, and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

g) using the new fit parameters Δt and δt, determining the net thickness of the reservoir by Δ_(iNET)=Δt−δt; and

h) using the gross thickness Δt and the net thickness of the reservoir Δ_(iNET)=Δt−δt, determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot \left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right)}} \end{matrix}$

depending on whether the data type is P-P, S-S or P-S respectively.

6. A method of determining the net thickness of a reservoir bed, comprising the following steps:

a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces;

b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui);

c) determining the frequency fx at which the amplitudes are at their maximum value by analyzing the amplitude frequency pairs A_(i) and f_(Ui);

d) using linear fitting methods, and using Δt=1/(1.4*fx), fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}} + {\frac{2\pi^{4}}{15}\left( {\delta \quad {tf}_{U}} \right)^{4}}} \right\rbrack}$

to generate fit parameters Δt, Σδt_(i), and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

e) substituting the fit parameter Σδt_(i) into ${{\Delta \quad t} = {\frac{5}{7{fx}} - ɛ}},$

where ε=2.14*ƒx*Σδt_(i) ², to produce an improved estimate of Δt;

f) using the improved estimate of Δt, and using linear fitting methods, fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}} + {\frac{2\pi^{4}}{15}\left( {\delta \quad {tf}_{U}} \right)^{4}}} \right\rbrack}$

to generate new fit parameters Δt, Σδt_(i), and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

g) using the new fit parameters Δt and Σδt_(i), determining the net thickness of the reservoir by ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad \underset{i}{t}}}}};$

h) using the gross thickness Δt and the net thickness of the reservoir ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad \underset{i}{t}}}}},$

determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot \left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right)}} \end{matrix}$

depending on whether the data type is P-P, S-S or P-S respectively.

7. A method of determining the net thickness of a reservoir bed, comprising the following steps:

a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces;

b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui);

a) in the case where only a single embedded layer of thickness δt is present, determining the frequency fx at which the amplitudes are at their maximum value by analyzing the amplitude frequency pairs A_(i) and f_(Ui);

d) using linear fitting methods, and using Δt=1/(1.4*fx), fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}} + {\frac{2\pi^{4}}{15}\left( {\delta \quad {tf}_{U}} \right)^{4}}} \right\rbrack}$

to generate fit parameters Δt, δt, and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

e) substituting the fit parameter δt into the following equation, ${{\Delta \quad t} = {\frac{5}{7{fx}} - ɛ}},$

where

ε=2.14*ƒx*δt²

to produce an improved estimate of Δt;

f) using the improved estimate of Δt, and using linear fitting methods, fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {\alpha \quad \Delta \quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}} + {\frac{2\pi^{4}}{15}\left( {\delta \quad {tf}_{U}} \right)^{4}}} \right\rbrack}$

to generate new fit parameters Δt, δt, and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

g) using the new fit parameters Δt and δt, determining the net thickness of the reservoir by Δ_(tNET)=Δt−δt; and

h) using the gross thickness Δt and the net thickness of the reservoir Δ_(tNET)=Δt−δt,

determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot \left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right)}} \end{matrix}$

depending on whether the data type is P-P, S-S or P-S respectively.

8. A method of determining the net thickness of a reservoir bed, comprising the following steps:

a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces;

b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui);

c) extracting two amplitude measurements fx and fh and their corresponding frequencies fx and fh from the amplitude frequency data where the first measurement is made where the amplitude is at its maximum (Ax, fx) and the second is for a frequency at least 10% higher than the frequency where the amplitude is at its maximum;

a) determining Σδτ, using the following equation, ${\sum{\delta \quad t}} = {(1.013)\sqrt{\frac{{C_{1}f_{x}{{SIN}\left\lbrack \frac{10\pi \quad f_{h}}{7f_{x}} \right\rbrack}} + {C_{2}f_{h}\frac{A_{h}}{A_{x}}} - {C_{3}f_{h}}}{{C_{4}f_{h}f_{x}^{2}} - {C_{5}f_{h}^{3}} - {C_{6}f_{x}^{3}{{SIN}\left\lbrack \frac{10\pi \quad f_{h}}{7f_{x}} \right\rbrack}}}}}$

e) determining Δt, using ${{\Delta \quad t} = {\frac{5}{7{fx}} - ɛ}},$

where

ε=2.14*ƒx*Εδt_(i) ²;

f) using Δt and Σδτ, determining the net thickness of the reservoir by using ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum{\delta \quad \underset{i}{t}}}}};$

and;

g) using the gross thickness Δt and the net thickness of the reservoir ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum{\delta \quad \underset{i}{t}}}}},$

determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot \left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right)}} \end{matrix}$

depending on whether the data type is P-P, S-S or P-S respectively.

9. A method of determining the net thickness of a reservoir bed, comprising the following steps:

a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui) to generate filtered seismic traces;

a) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui);

a) in the case where only a single embedded layer of thickness δt is present, extracting two amplitude measurements fx and fh and their corresponding frequencies fx and fh from the amplitude frequency data where the first measurement is made where the amplitude is at its maximum (Ax, fx) and the second is for a frequency at least 10% higher than the frequency where the amplitude is at its maximum;

c) determining δτ, using ${\delta \quad t} = {(1.013)\sqrt{\frac{{C_{1}f_{x}{{SIN}\left\lbrack \frac{10\pi \quad f_{h}}{7f_{x}} \right\rbrack}} + {C_{2}f_{h}\frac{A_{h}}{A_{x}}} - {C_{3}f_{h}}}{{C_{4}f_{h}f_{x}^{2}} - {C_{5}f_{h}^{3}} - {C_{6}f_{x}^{3}{{SIN}\left\lbrack \frac{10\pi \quad f_{h}}{7f_{x}} \right\rbrack}}}}}$

d) determining Δt, using ${{\Delta \quad t} = {\frac{5}{7{fx}} - ɛ}},$

where

ε=2.14*ƒx*δt²

e) using Δt and δτ, determining the net thickness of the reservoir by using Δ_(iNet)=Δt−δt; and

i) using the gross thickness Δt and the net thickness of the reservoir Δ_(iNET)=Δt−δt,

determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot \left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right)}} \end{matrix}$

depending on whether the data type is P-P, S-S or P-S respectively.

Although this detailed description has shown and described illustrative embodiments of the invention, this description contemplates a wide range of modifications, changes, and substitutions. In some instances, one may employ some features of the present invention without a corresponding use of the other features. Accordingly, it is appropriate that readers should construe the appended claims broadly, and in a manner consistent with the scope of the invention. 

What is claimed is:
 1. A method of determining the net thickness of a reservoir bed, comprising the following steps: a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui) to generate filtered seismic traces; b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui); c) determining the gross reservoir thickness Δt and the thickness of the non-reservoir material Σδ₁, using the extracted amplitude frequency pairs A_(i) and the f_(Ui); and d) determining the net thickness of the reservoir by $\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad {\underset{i}{t}.}}}}$


2. A method of determining the net thickness of a reservoir bed, comprising the following steps: a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces; b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui); c) fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, using nonlinear fitting methods, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {{\alpha\Delta}\quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad t\quad f_{U}}} \right) - {\sum\limits_{i}\left( {1 - \frac{{SIN}\left( {{\alpha\delta}\quad t_{i}f_{U}} \right)}{{\alpha\delta}\quad t_{i}f_{U}}} \right)}} \right\rbrack}$

to generate fit parameters Δt, Σδt₁, and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

d) using the fit parameters Δt and Σδt₁, determining the net thickness of the reservoir by using ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad {\underset{i}{t}.}}}}};$

and e) using the gross thickness Δt and the net thickness of the reservoir ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad {\underset{i}{t}.}}}}},$

determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot {\left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right).}}} \end{matrix}$


3. A method of determining the net thickness of a reservoir bed, comprising the following steps: a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces; b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui); c) in the case where only a single embedded layer of thickness δt is present, using nonlinear fitting methods, fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {{\alpha\Delta}\quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad t\quad f_{U}}} \right) - \left( {1 - \frac{{SIN}\left( {{\alpha\delta}\quad {tf}_{U}} \right)}{{\alpha\delta}\quad {tf}_{U}}} \right)} \right\rbrack}$

to generate fit parameters Δt, δt, and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

d) using the fit parameters Δt and δt, determining the net thickness of the reservoir by using Δ_(iNET)=Δt−δt; and e) using the gross thickness Δt and the net thickness of the reservoir Δ_(iNET)=Δt−δt, determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot {\left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right).}}} \end{matrix}$


4. A method of determining the net thickness of a reservoir bed, comprising the following steps: a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces; b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui); c) determining the frequency fx at which the amplitudes are at their maximum value by analyzing the amplitude frequency pairs A_(i) and f_(I); d) using linear fitting methods, and using Δt=1/(1.4*fx), fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {{\alpha\Delta}\quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad t\quad f_{U}}} \right) - {\sum\limits_{i}{\frac{2\pi^{2}}{3}\left( {\delta \quad t_{i}f_{U}} \right)^{2}}}} \right\rbrack}$

to generate fit parameters Δt, Σδt_(i), and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

e) substituting the fit parameter Σδt_(i) into ${\Delta \quad t} = {{\frac{5}{7\quad {fx}} - {ɛ\quad {where}\quad ɛ}} = {2.14*{fx}*{\sum{\delta \quad t_{i}^{2}}}}}$

to produce an improved estimate of Δt; f) fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, using linear fitting methods, and using the improved estimate of Δt, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {{\alpha\Delta}\quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad t\quad f_{U}}} \right) - {\sum\limits_{i}{\frac{2\pi^{2}}{3}\left( {\delta \quad t_{i}f_{U}} \right)^{2}}}} \right\rbrack}$

to generate new fit parameters Δt, Σδt_(i), and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

g) using the new fit parameters Δt and Σδt_(i), determining the net thickness of the reservoir by ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad \underset{i}{t}}}}};$

and h) using the gross thickness Δt and the net thickness of the reservoir ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad \underset{i}{t}}}}},$

determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot {\left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right).}}} \end{matrix}$


5. A method of determining the net thickness of a reservoir bed, comprising the following steps: a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces; b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui); c) in the case where only a single embedded layer of thickness δt is present, determining the frequency fx at which the amplitudes are at their maximum value by analyzing the amplitude frequency pairs A_(i) and f_(Ui); d) using linear fitting methods, and using Δt=1/(1.4*fx), fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {{\alpha\Delta}\quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad t\quad f_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}}} \right\rbrack}$

to generate fit parameters Δt, δt, and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

e) substituting the fit parameter δt into the following equation, ${\Delta \quad t} = {\frac{5}{7{fx}} - ɛ}$

where ε=2.14*ƒx*δt² to produce an improved estimate of Δt; f) using the improved estimate of Δt, and using linear fitting methods, fitting the amplitude and frequency pairs A_(i) and f_(Ui), to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {{\alpha\Delta}\quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad t\quad f_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}}} \right\rbrack}$

to generate new fit parameters Δt, δt, $\frac{\Delta \quad Z_{p}}{Z_{p}};$

g) using the new fit parameters Δt and δt, determining the net thickness of the reservoir by Δ_(iNET)=Δt−δt; and h) using the gross thickness Δt and the net thickness of the reservoir Δ_(iNET)=Δt−δt, determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot {\left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right).}}} \end{matrix}$


6. A method of determining the net thickness of a reservoir bed, comprising the following steps: a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces; b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui); c) determining the frequency fx at which the amplitudes are at their maximum value by analyzing the amplitude frequency pairs A_(i) and f_(Ui); d) using linear fitting methods, and using Δt=1/(1.4*fx), fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {{\alpha\Delta}\quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}} + {\frac{2\pi^{4}}{15}\left( {\delta \quad {tf}_{U}} \right)^{4}}} \right\rbrack}$

to generate fit parameters Δt, Σδt_(i), and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

e) substituting the fit parameter Σδt_(i) into ${{\Delta \quad t} = {\frac{5}{7{fx}} - ɛ}},$

where ε=2.14*ƒx*Σδt_(i) ², to produce an improved estimate of Δt; f) using the improved estimate of Δt, and using linear fitting methods, fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {{\alpha\Delta}\quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}} + {\frac{2\pi^{4}}{15}\left( {\delta \quad {tf}_{U}} \right)^{4}}} \right\rbrack}$

to generate new fit parameters Δt, Σδt_(i), and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

g) using the new fit parameters Δt and Σδt_(i), determining the net thickness of the reservoir by ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad \underset{i}{t}}}}};$

and h) using the gross thickness Δt and the net thickness of the reservoir ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum\limits_{i}{\delta \quad \underset{i}{t}}}}},$

determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot {\left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right).}}} \end{matrix}$


7. A method of determining the net thickness of a reservoir bed, comprising the following steps: a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces; b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui); c) in the case where only a single embedded layer of thickness δt is present, determining the frequency fx at which the amplitudes are at their maximum value by analyzing the amplitude frequency pairs A_(i) and f_(Ui); d) using linear fitting methods, and using Δt=1/(1.4*fx), fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\quad \overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {{\alpha\Delta}\quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad t\quad f_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}} + {\frac{2\pi^{4}}{15}\left( {\delta \quad t\quad f_{U}} \right)^{4}}} \right\rbrack}$

to generate fit parameters Δt, δt, and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

e) substituting the fit parameter δt into the following equation, ${{\Delta \quad t} = {\frac{5}{7{fx}} - ɛ}},$

where ε=2.14*ƒx*δt² to produce an improved estimate of Δt; f) using the improved estimate of Δt, and using linear fitting methods, fitting the amplitude and frequency pairs A_(i) and f_(Ui) to the following equation, ${{AMP}\left( f_{U} \right)} \cong {\frac{\Delta \quad {Zp}}{2\overset{\_}{Zp}}\left\lbrack {\left( {1 - \frac{{SIN}\left( {{\alpha\Delta}\quad {tf}_{U}} \right)}{\alpha \quad \Delta \quad {tf}_{U}}} \right) - {\frac{2\pi^{2}}{3}\left( {\delta \quad {tf}_{U}} \right)^{2}} + {\frac{2\pi^{4}}{15}\left( {\delta \quad {tf}_{U}} \right)^{4}}} \right\rbrack}$

to generate new fit parameters Δt, δt, and $\frac{\Delta \quad Z_{p}}{Z_{p}};$

g) using the new fit parameters Δt and δt, determining the net thickness of the reservoir by Δ_(iNET)=Δt−δt; and h) using the gross thickness Δt and the net thickness of the reservoir Δ_(iNET)=Δt−δt, determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot {\left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right).}}} \end{matrix}$


8. A method of determining the net thickness of a reservoir bed, comprising the following steps: a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui), to generate filtered seismic traces; b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui); c) extracting two amplitude measurements fx and fh and their corresponding frequencies fx and fh from the amplitude frequency data where the first measurement is made where the amplitude is at its maximum (Ax, fx) and the second is for a frequency at least 10% higher than the frequency where the amplitude is at its maximum; d) determining Σδτ, using the following equation, ${\sum{\delta \quad t}} = {(1.013)\sqrt{\frac{{C_{1}f_{x}{{SIN}\left\lbrack \frac{10\pi \quad f_{h}}{7f_{x}} \right\rbrack}} + {C_{2}f_{h}\frac{A_{h}}{A_{x}}} - {C_{3}f_{h}}}{{C_{4}f_{h}f_{x}^{2}} - {C_{5}f_{h}^{3}} - {C_{6}f_{x}^{3}{{SIN}\left\lbrack \frac{10\pi \quad f_{h}}{7f_{x}} \right\rbrack}}}}}$

e) determining Δt, using ${{\Delta \quad t} = {\frac{5}{7{fx}} - ɛ}},$

where ε=2.14* ƒx*Σδt_(i) ²; f) using Δt and Σδτ, determining the net thickness of the reservoir by using ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum{\delta \quad \underset{i}{t}}}}};$

and g) using the gross thickness Δt and the net thickness of the reservoir ${\Delta_{t_{NET}} = {{\Delta \quad t} - {\sum{\delta \quad \underset{i}{t}}}}},$

determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot {\left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right).}}} \end{matrix}$


9. A method of determining the net thickness of a reservoir bed, comprising the following steps: a) filtering a set of seismic data with a plurality of band pass filters, wherein the set is selected from the group consisting of normal moveout corrected and offset stacked seismic traces, or angle stacked seismic traces, with each filter designed to have a common lower frequency f_(L), and progressively increasing upper frequency f_(Ui) to generate filtered seismic traces; b) extracting amplitudes from the group consisting of the filtered seismic traces or the energy envelope of the filtered seismic traces, using a method selected from the group consisting of either a sample-by-sample method or a lobe method, wherein the extracted amplitudes and their corresponding frequencies are A_(i) and the f_(Ui); c) in the case where only a single embedded layer of thickness δt is present, extracting two amplitude measurements fx and fh and their corresponding frequencies fx and fh from the amplitude frequency data where the first measurement is made where the amplitude is at its maximum (Ax, fx) and the second is for a frequency at least 10% higher than the frequency where the amplitude is at its maximum; d) determining δτ, using ${\delta \quad t} = {(1.013)\sqrt{\frac{{C_{1}f_{x}{{SIN}\left\lbrack \frac{10\pi \quad f_{h}}{7f_{x}} \right\rbrack}} + {C_{2}f_{h}\frac{A_{h}}{A_{x}}} - {C_{3}f_{h}}}{{C_{4}f_{h}f_{x}^{2}} - {C_{5}f_{h}^{3}} - {C_{6}f_{x}^{3}{{SIN}\left\lbrack \frac{10\pi \quad f_{h}}{7f_{x}} \right\rbrack}}}}}$

e) determining Δt, using ${{\Delta \quad t} = {\frac{5}{7{fx}} - ɛ}},$

where ε=2.14*ƒx*δt² f) using Δt and δτ, determining the net thickness of the reservoir by using Δ_(iNET)=Δt−δt; and g) using the gross thickness Δt and the net thickness of the reservoir Δ_(iNET)=Δt−δt, determining the spatial thickness ΔZ and ΔZ_(NET) using an equation from the group consisting of the following three equations: $\begin{matrix} \begin{matrix} {{\Delta \quad Z} = {\frac{\Delta \quad T_{{pp} - {2{WAY}}}}{2} \cdot {Vp}_{INT}}} \\ {{\Delta \quad Z} = {\frac{\Delta \quad T_{{ss} - {2{WAY}}}}{2} \cdot {Vs}_{INT}}} \\ {{\Delta \quad Z} = {T_{PS} \cdot {\left( \frac{{Vp} \cdot {Vs}}{{Vp} + {Vs}} \right).}}} \end{matrix} & (1) \end{matrix}$ 